Happy holidays. I’m presenting another article by drew hempel. I’ve asked drew to write in more detail on the subject of the connection between math and music and how it connects to the paranormal. I felt that his article
The Secret of Psychic Music Healing was assuming too much of the reader and so asked him to write a more explanatory article, which I now present.


Against Archytas:  How the West Lost Alchemy or Paranormal Complimentary Opposite Harmonics

by drew hempel, MA

(anti-copyright, free distribution)

The early Greek mathematics used the 60-based number system of Babylon from which Archytas, a collaborator with Plato, received the harmonic tetrachord or the continued proportion 6:8::9:12.  This tetrachord creates a geometric mean between the octave, perfect fourth and perfect fifth music intervals, or 1:2:3:4, through “divide and average” logarithmic-based mathematics.  So 6:8 and 9:12 are in the continued proportion 3:4, the perfect fourth music interval, while 6:9 and 8:12 are 2:3, the perfect fifth music interval, and 6:12 is 1:2, the octave.  The geometric mean is A:B::B:C or B squared = AC or the square root of AC = B. What Archytas added to this Babylon “divide and average” harmonic mathematics was the concept of the Greek “incommensurable” – the algebraic axiomatic proof of “alogon” or a precise irrational number, the square root of two.  This process ushered in what’s called “The Greek Miracle” that continues to be the structure of science:  symmetry-based mathematics.

Instead of the above system, the alchemical Pythagorean Tetrad relies on complimentary opposite harmonics so that an equilateral triangle of geometric points equals the continued proportion 1:2:3:4 as the octave, perfect fifth and perfect fourth music intervals.   In “orthodox” Pythagorean harmonics this was known as the “subcontrary mean” whereby the complimentary opposites of the Tetrad were maintained in violation of “divide and average” mathematics.  So for the Tetrad A:B is 2:3 and B:A is 3:4 against the commutative property, A x B = B x A.   In music theory this complimentary opposite inversion of the perfect fifth and perfect fourth is taught as 2:3 is C to G while 3:4 is G to C.  This process of complimentary opposites is listened to, as the perfect fifth, perfect fourth harmonics, which create all the notes.  Most importantly the complimentary opposite harmonics transduces sound throughout the whole energy spectrum, as I’ve described in previous articles..   

Philolaus, one of the early Pythagorean writers, detailed that this “subcontrary mean” or complimentary opposite harmonic caused any attempt at subdividing the scale into symmetry as a failure.  In contrast Archytas changed the “subcontrary” complimentary opposite mean into the “harmonic mean” using “divide and average” mathematics.  The outcome has precisely opposite the meaning of “harmony” which for Pythagoreans referred to the paranormal source of sound as the Goddess Harmonia or what I call female formless awareness.  For Philolaus the perfect fifth as 2:3 could be inverted to 3:2 and then extended another fifth to 9:4 and then divided back into the octave, below 2, for the major second interval of 9:8 or C to D.  Yet 9:8 cubed or three major second music intervals equaled the 3:2 perfect fifth music interval, plus a tiny ratio called “the comma of Pythagoras.”  

This “comma of Pythagoras” is the difference between the “divide and average” octave system adopted by Archytas and the complimentary opposite fifths inverting into fourths, used by the orthodox Pythagoreans.  The “comma of Pythagoras” is the key to harmonic alchemy whereby 2:3, the perfect fifth, is yang in Taoism and 3:4, the perfect fourth, is yin.  As Gurdjieff desribes the “shock” of the diatonic scale, whereby the “inverse ratios” do not line up with the octaves, is intensified as the octaves expand.  In contrast Archytas argued that 9/8 cubed or three major second intervals equals the square root of two as the Greek Miracle, the axiomatic algebra of the precise incommensurable irrational number.  What Archtyas essentially did, as I’ll describe, is equate the perfect 5th or 2:3 with the perfect 4th or 3:4 as equally-divided or symmetric ratios through a “divide and average” mathematics.

Archytas took the Babylonian geometric mean of 6:8::9:12 used for harmonics and then applied the Pythagorean Tetrad 1:2:3:4 so that the 2:3 ratio of complimetary opposite frequency was converted to 3:2 as a materialistic vibrating string length.  This became known as the Law of Pythagoras even though it goes against the true meaning of the complimentary opposites when this “inverse ratio” is combined with the “divide and average” commutative property. Gurdjieff, for example, still relies on the “inverse ratio” of density or string length versus frequency or consciousness.  But Gurdjieff does not use Archytas’ “divide and average” symmetric-based mathematics, instead Gurdjieff relies on the Law of Three aka the Tetrad, or octave-fifth-fourth,  to resonate through the comma of Pythagoras as the “shocks” of alchemy.  In Taoism this system of alchemical shocks is taught as the 12 harmonic nodes along the outside of the body, enabling healing and paranormal energy, an exercise called “the small universe.”

The equation used by Archtyas, from Babylon, was arthimetic mean times harmonic mean = geometric mean squared.  For the Pythagorean Tetrad this means if A = 1 and C = 2, the octave, then B = 3/2 for the arithmetic mean (A + C divided by 2) and B = 4/3 for the harmonic mean or 2(AC) divided by A + C. Meanwhile B = the square root of two for the geometric mean or the arithmetic mean times the harmonic mean equals the geometric mean squared (3/2×4/3 = 2).

The error that Archytas makes is ignoring the “comma of Pythagoras” arising from complimentary opposites harmonics so that the square root of two is equated with 3/2 through the “divide and average” converging sequence.  The square root of two is greater then one and less than two which can be solved through the “divide and average” geometric-based continued fraction series of (1 + a)squared = 1 + 2a + a squared.  2a + a squared = 1.  Or a(2 + a)=1.  Therefore a = 1 divided by 2 + a.  Then just keep replacing “a” with “1 divided by 2 + a” for an infinite series that converges as the square root of two.  The first iteration is 1 + 1/2 or the approximate series solution, 3/2 = the square root of two.

It needs to be emphasized that the square root of two is actually a “transcendental” number although this is only acknowledged in number theory and not in standard math.  There is no positive proof demonstrating that the hypotenuse of the 1-1 triangle equals the algebraic symbol the square root of two since there’s no algebric equation demonstrating the square root of two, only an infinite algebraic series that is not a closed set.  Most people just learn the geometric proof for the square root of two, demonstrating a “proof by contradiction,” which simply states that the ratios are not rational, but it does not consider other options for the ratios, especially the case of diverging complimentary opposite harmonics.

In contrast music theory, based on logical inference, uses listening to complimentary opposites, 1:2:3:4, as proof that the foundation of reality is female consciousness with the male number 1 as an infinite resonance creating an octave, 1/2, that expands in frequency through the perfect fifth or 2:3 or yang, inverting to the perfect fourth or 3:4 as yin.  In western music theory this is taught as the “circle of fifths” so that the perfect fifth or 2:3 starts with C to G then continues to finish the octave C through the 12 notes of the scale.  C-G-D-A-E-B-F#-C#-G#-D#-A#-F-C.  This return back to “C” again ignores the empirical fact that the perfect fifth overtone inverts into the perfect fourth interval – through complimentary opposites, a dynamic that can be heard in music.  So if you have a string marked  into 1/2, 1, 3/2 and 2, a node at 3/2 actually is 3/4 of the string.  Or you could start the string with zero, and again a node at 2/3 is 3/4 of the string.  Similarly a node at 3/4 is 2/3 of the string.  What’s important is the yin-yang dynamic of this complimentary opposite harmonic thereby creating alchemy.

In other words there’s a complimentary opposite between the nodes and the numbers which can be heard as harmonious.  For this reason that musical “inverse induction” extension of the perfect fifths does not line up with the the extension of the octaves and this difference is called the “comma of Pythagoras.” In Nature there is no “circle of fifths,” but rather an infinite spiral of energy transduction that starts and ends with the male number one resonating as a complimentary opposite into female formless awareness and then resonating out as the perfect fifth harmonic which inverts as the perfect fourth to pull back to one and then, through its complimentary opposite – female formless awareness.  This process continues as resonance overtones, just as Dennis Gabor’s Quantum Time-Frequency Uncertainty Principle describes:  As the time gets less the frequency spreads across the whole energy spectrum.  It’s no accident that in quantum mechanics the commutative property is also violated with momentum times position not equaling position times momentum – only in science this is converted back into symmetric-based math using logarithms.  Quantum physicist Henry P. Stapp makes this paradox of Number and Order central to the mystery of consciousness in quantum mechanics.

The Golden Ratio also does not converge geometrically, unless, like the perfect fifth music interval, 2:3, the order of the Fibonacci number series is reversed so that there is symmetry between zero and one, using the “divide and average” commutative property.  In the converging series expansion the continued proportion A:B::B:A + B of the Fibonacci series, 1, 2, 3, 5, 8, 13 is reversed to A:B::B:A-B so that A(A – B) gives a positive solution for geometric convergence.  What this means, again, in terms of music is that the frequency ratios of 4:5, the major third, and 5:8, the minor sixth have to be reversed to materialistic string length as 5:4, the cube root of two, and 8:5, an approximation for the golden ratio and also the cube root of four.  Kepler used this Golden Ratio as the 3:5 music harmonic to combine it with 5:4, the major third, in order to justify his elliptical orbit analysis.  Similarly Newton’s inverse square law of gravity was derived from his application of Archytas’ symmetry-based logic so that it takes four times the weight on the end of a string to increase the tension to twice the string’s frequency.  Not until the 1960s did the Philosophical Transactions of the Royal Society of London publish this newly discovered Pythagorean-based source for Newton’s gravity.

I discovered that Archytas’ proof for precisely doubling the cube, his most famous equation, relies on this secret harmonic conversion of ratios into irrationals.  Up until now, only a geometric magnitude proof is given for doubling the cube so that A:X::Y:2A refers to X being the side length of the cube that needs to be doubled while the volume as 2A needs to be cubed with the geometric mean equation as 1:2::4:8.  The original side length equation is 1:cube root of two::cube root of four:2.  In fact these geometric mean ratios are from music harmonics as the major third and its inverse, the minor sixth or Golden Ratio, giving 1:5/4::8/5:2.  None of the mathematicians have been able to make this connection to Archytas’ conversion of the 2:3, perfect fifth music interval, for this “doubling the cube” proof that creates the Greek Miracle – an axiom for an irrational number.  David Fowler’s Mathematics of Plato’s Academy book certainly acknowledges the mystery of the music ratios origin for Archtyas’ doubling the cube proof.  Other mathematicians, like Professor Luigi Borzacchini, with whom I’ve corresponded, have certainly pondered the math-music incommensurability issue.

And so again the continued proportion proof for the square root of two in its first iteration gives the value 1 + 1/2 or 3/2 = the square root of two which is what Archytas relied on.  This same proof is tested by squaring both sides so that 9/4 = 2 (a first iteration in the series) or algebraically y squared + or – 1 divided by x squared = 2.  In the geometric mean proof, also used for the Golden Ratio, AC = B squared or N = x(x-1).  This also converts back to the Babylonian geometric mean equation so that 6:8::9:12.  Because 9/4 = 2, as the first iteration of the square root of two series, this is actually based on the Tetrad harmonics of 3/2 squared which must be divided back into the octave, less than two, as 9/8, the major second interval.  Or again Archytas combined the complimentary opposite harmonics of 1:2:3:4 into the geometric mean 6:8::9:12 for an algebraic solution to the square root of two with the iteration of 3:2.  So the geometric mean became paradoxically 1:8::9:2 or 1:2 cubed::3 squared:2 but the order is reversed to 9/8 since it’s a materialist string length, and not just frequency ratios.  In terms of the geometric mean for doubling the cube we can see that 5/4 is just 9/8 as two major second intervals or 10/8 reduced to 5/4, now justified as the cube root of two as the major third while the minor sixth, as 8:5, is just the half and then inverted.   

Again Archytas’ use of “just tuning” diatonic ratios, also used by Gurdjieff, such as 9:8, 3:5, and 5:4 were solely due to Archtyas’ conversion of 2:3:4 as complimentary opposites into the “divide and average” symmetry or the arithmetic mean.  So 2:3 became 3:2 from A + C divided by 2 = B with A = 1 and C = 2 as the octave.  This conversion of the complimentary opposite Tetrad perfect fifth of 2:3 into 3:2, as an arithmetic mean, then could be doubled, 9/4, and inverted back into the octave as 9/8, the major second interval and then converted to the geometric mean as three major second intervals or the square root of two also known as the “Devil’s Interval,” the tritone – C to F#.  

An easier way to understand this is described on this physics website:  http://hypertextbook.com/physics/waves/music/

“Solution … 
The ratio of the diagonal of a square to a side is √2:1. (Galileo stated the order of the ratio the other way around, but this is a minor detail.) Each half step (a semitone) up the equal tempered scale multiplies the previous note by the twelfth root of two, two half steps (a whole tone) multiplies the note by the twelfth root of two squared, three half steps by the twelfth root of two cubed, and so on …

1 semitone   minor second 12√2 12√2 
2 semitones  = 1 tone (whole tone)  major second 12√2 12√2 6√2 
3 semitones   minor third 12√2 12√2 12√2 4√2 
4 semitones  = 2 tones (ditone) major third 12√2 12√2 12√2 12√2 3√2 
5 semitones   perfect fourth 12√2 12√2 12√2 12√2 12√2 2.4√2 
6 semitones  = 3 tones (tritone) augmented fourth  12√2 12√2 12√2 12√2 12√2 12√2 2√2

Six semitones is equal to the twelfth root of two to the sixth power, which is equal to the square root of two. This interval is called a tritone, an augmented fourth, or a diminished fifth; for example, C and F♯ (G♭) or F and B. Had I given you a more complete quote from Galileo you would have already known this.”

In orthodox Pythagoreanism this use of an attempt to equal-temper the scale into 12 fifths, using the 9/8 ratio as the geometric mean, would not have been allowed since it enables setting up a quadratic equation based on the commutative property and the Pythagorean Theorem.  A square x B squared = C squared was always averaged in other mathematic systems but now it could be combined with zero to create a geometric convergence as an irrational number – the Greek Miracle based on a deep disharmony.  The complimentary opposite harmonics which naturally resonate as yin-yang dynamics have now been destroyed.  Even supersymmetry is just an extension of what Professor Oliver Reiser called “the music logarithmic spiral” so that mass squared is inversely proportional to energy frequency distance, as detailed by physics professor Gordon Kane.

There is no pure science.  Archytas’ “doubling the of cube” miracle was used for catapult technololgy while Newton’s “inverse square law” from Archytas’ geometric mean analysis has been the key for all projectile military technology just as Galileo’s Pythaogorean math was used for cannons.

What enabled Archytas to ignore the “comma of Pythagoras” which in alchemy creates an infinite resonance of energy through complimentary opposites was the vast difference of error between the Golden Ratio, the slowest converging irrational number and the square root of two series.  So that the inversion of 5:8 into 8:5 for both string length use (instead of frequency) and for use of the quadratic zero, geometric convergence, does not greatly affect the harmonic series accuracy.  It’s inverse is the conversion of 5/4 or 10/8 into the cube root of two.  This is the true secret of the Greek Miracle – combining the Brahmin cipher system of the commutative property, using zero, with Babylon’s equation of the arithmetic mean times harmonic mean equaling the geometric mean squared.  Eudoxus simply extended Archtyas’ use of the harmonic mean so that it could be applied as the Golden Ratio mean.  As the book Excursions into Number Theory notes:  “For instance, 99/70, the sixth convergent of square root of two, differs from square root of two by .000072; but 13/8, the sixth convergence of the Golden Ratio, differs from the Golden Ratio by .0070, showing an error nearly 100 times as large.” (p. 134)

In conclusion – mathematicians have never been able to figure out why the ratio 2/3 was sacred in Egypt.  Now you know the secret – or at least to what extent complimentary opposite alchemical harmonics have been covered up.